Proposition

Let (C,t)(C,t) be an ∞\infty-site and f:F→Gf : F \to G a morphism of presheaves in P(C)P(C). The morphism at(f)a_t(f) is an effective epimorphism in Shvt(C)Shv_t(C) if and only if ff is a local epimorphism, i.e.

Example

If S1=*∐*∐**S^1 = \ast \underset{\ast \coprod \ast}{\coprod} \ast denotes the homotopy type of the circle, then the unique morphism S1→Δ0S^1 \to \Delta^0 is an effective epimorphism, by prop. 7, but it not an epimorphism, because the suspension of S1S^1 is the sphereS2S^2, which is not contractible.